coefficients of equivariant complex cobordism

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Coefficients of equivariant complex cobordism Yunze Lu University of Michigan August, 2019 1 / 24 Complex cobordism Complex manifolds: Compact smooth manifolds, with a tangential stable almost complex structure. Two closed manifolds are

Coefficients of equivariant complex cobordism Yunze Lu University of Michigan August, 2019 1 / 24
Complex cobordism Complex manifolds: Compact smooth manifolds, with a tangential stable almost complex structure. Two closed manifolds are cobordant, if their disjoint union is the boundary of a third manifold. This is an equivalent relation. Complex cobordism ring Ω U ∗ (graded), under disjoint union and Cartesian product. 2 / 24
Complex cobordism Complex manifolds: Compact smooth manifolds, with a tangential stable almost complex structure. Two closed manifolds are cobordant, if their disjoint union is the boundary of a third manifold. This is an equivalent relation. Complex cobordism ring Ω U ∗ (graded), under disjoint union and Cartesian product. 2 / 24
Complex cobordism Complex manifolds: Compact smooth manifolds, with a tangential stable almost complex structure. Two closed manifolds are cobordant, if their disjoint union is the boundary of a third manifold. This is an equivalent relation. Complex cobordism ring Ω U ∗ (graded), under disjoint union and Cartesian product. 2 / 24
Complex cobordism Complex manifolds: Compact smooth manifolds, with a tangential stable almost complex structure. Two closed manifolds are cobordant, if their disjoint union is the boundary of a third manifold. This is an equivalent relation. Complex cobordism ring Ω U ∗ (graded), under disjoint union and Cartesian product. 2 / 24
Thom’s Theorem Thom space Th ( ξ ) . Universal n -complex bundle γ n . Thom’s homomorphism: τ : π k + 2 n Th ( γ n ) → Ω U k . Theorem (Thom, 54) τ is an isomorphism for large n . Those Thom spaces could be assembled to form a spectrum called ∗ ∼ MU , and Ω U = π ∗ MU . Theorem (Milnor, Novikov, 60) MU ∗ = π ∗ MU = Z [ x 1 , x 2 , ... ] where x i ∈ π 2 i MU . 3 / 24
Thom’s Theorem Thom space Th ( ξ ) . Universal n -complex bundle γ n . Thom’s homomorphism: τ : π k + 2 n Th ( γ n ) → Ω U k . Theorem (Thom, 54) τ is an isomorphism for large n . Those Thom spaces could be assembled to form a spectrum called ∗ ∼ MU , and Ω U = π ∗ MU . Theorem (Milnor, Novikov, 60) MU ∗ = π ∗ MU = Z [ x 1 , x 2 , ... ] where x i ∈ π 2 i MU . 3 / 24
Thom’s Theorem Thom space Th ( ξ ) . Universal n -complex bundle γ n . Thom’s homomorphism: τ : π k + 2 n Th ( γ n ) → Ω U k . Theorem (Thom, 54) τ is an isomorphism for large n . Those Thom spaces could be assembled to form a spectrum called ∗ ∼ MU , and Ω U = π ∗ MU . Theorem (Milnor, Novikov, 60) MU ∗ = π ∗ MU = Z [ x 1 , x 2 , ... ] where x i ∈ π 2 i MU . 3 / 24
Thom’s Theorem Thom space Th ( ξ ) . Universal n -complex bundle γ n . Thom’s homomorphism: τ : π k + 2 n Th ( γ n ) → Ω U k . Theorem (Thom, 54) τ is an isomorphism for large n . Those Thom spaces could be assembled to form a spectrum called ∗ ∼ MU , and Ω U = π ∗ MU . Theorem (Milnor, Novikov, 60) MU ∗ = π ∗ MU = Z [ x 1 , x 2 , ... ] where x i ∈ π 2 i MU . 3 / 24
Thom’s Theorem Thom space Th ( ξ ) . Universal n -complex bundle γ n . Thom’s homomorphism: τ : π k + 2 n Th ( γ n ) → Ω U k . Theorem (Thom, 54) τ is an isomorphism for large n . Those Thom spaces could be assembled to form a spectrum called ∗ ∼ MU , and Ω U = π ∗ MU . Theorem (Milnor, Novikov, 60) MU ∗ = π ∗ MU = Z [ x 1 , x 2 , ... ] where x i ∈ π 2 i MU . 3 / 24
Thom’s Theorem Thom space Th ( ξ ) . Universal n -complex bundle γ n . Thom’s homomorphism: τ : π k + 2 n Th ( γ n ) → Ω U k . Theorem (Thom, 54) τ is an isomorphism for large n . Those Thom spaces could be assembled to form a spectrum called ∗ ∼ MU , and Ω U = π ∗ MU . Theorem (Milnor, Novikov, 60) MU ∗ = π ∗ MU = Z [ x 1 , x 2 , ... ] where x i ∈ π 2 i MU . 3 / 24
Homotopical equivariant complex cobordism MU G Compact Lie group G . Complete universe U . B U ( n ) : G -space of n -dimensional complex subspaces of U . Universal n -complex G -vector bundle γ n G . Complex finite dimensional representation V : G -vector bundle over a point. 4 / 24
Homotopical equivariant complex cobordism MU G Compact Lie group G . Complete universe U . B U ( n ) : G -space of n -dimensional complex subspaces of U . Universal n -complex G -vector bundle γ n G . Complex finite dimensional representation V : G -vector bundle over a point. 4 / 24
Homotopical equivariant complex cobordism MU G Compact Lie group G . Complete universe U . B U ( n ) : G -space of n -dimensional complex subspaces of U . Universal n -complex G -vector bundle γ n G . Complex finite dimensional representation V : G -vector bundle over a point. 4 / 24
Homotopical equivariant complex cobordism MU G Compact Lie group G . Complete universe U . B U ( n ) : G -space of n -dimensional complex subspaces of U . Universal n -complex G -vector bundle γ n G . Complex finite dimensional representation V : G -vector bundle over a point. 4 / 24
Homotopical equivariant complex cobordism MU G Compact Lie group G . Complete universe U . B U ( n ) : G -space of n -dimensional complex subspaces of U . Universal n -complex G -vector bundle γ n G . Complex finite dimensional representation V : G -vector bundle over a point. 4 / 24
Homotopical equivariant complex cobordism MU G Construction (tom Dieck, 70) For V ⊂ W , there are classifying map ( W − V ) × γ | V | → γ | W | G . G We have Th (( W − V ) × γ | V | = Σ W − V Th ( γ | V | G ) → Th ( γ | W | G ) ∼ G ) . Let D V = Th ( γ | V | G ) with the structured maps described above, then spectrify to obtain MU G . MU G is a genuine multiplicative G -specturm. It is complex stable: ( S V ∧ X ) . = MU ∗ + 2 | V | G ( X ) ∼ MU ∗ G 5 / 24
Homotopical equivariant complex cobordism MU G Construction (tom Dieck, 70) For V ⊂ W , there are classifying map ( W − V ) × γ | V | → γ | W | G . G We have Th (( W − V ) × γ | V | = Σ W − V Th ( γ | V | G ) → Th ( γ | W | G ) ∼ G ) . Let D V = Th ( γ | V | G ) with the structured maps described above, then spectrify to obtain MU G . MU G is a genuine multiplicative G -specturm. It is complex stable: ( S V ∧ X ) . = MU ∗ + 2 | V | G ( X ) ∼ MU ∗ G 5 / 24
Homotopical equivariant complex cobordism MU G Construction (tom Dieck, 70) For V ⊂ W , there are classifying map ( W − V ) × γ | V | → γ | W | G . G We have Th (( W − V ) × γ | V | = Σ W − V Th ( γ | V | G ) → Th ( γ | W | G ) ∼ G ) . Let D V = Th ( γ | V | G ) with the structured maps described above, then spectrify to obtain MU G . MU G is a genuine multiplicative G -specturm. It is complex stable: ( S V ∧ X ) . = MU ∗ + 2 | V | G ( X ) ∼ MU ∗ G 5 / 24
Homotopical equivariant complex cobordism MU G Construction (tom Dieck, 70) For V ⊂ W , there are classifying map ( W − V ) × γ | V | → γ | W | G . G We have Th (( W − V ) × γ | V | = Σ W − V Th ( γ | V | G ) → Th ( γ | W | G ) ∼ G ) . Let D V = Th ( γ | V | G ) with the structured maps described above, then spectrify to obtain MU G . MU G is a genuine multiplicative G -specturm. It is complex stable: ( S V ∧ X ) . = MU ∗ + 2 | V | G ( X ) ∼ MU ∗ G 5 / 24
Geometric equivariant complex cobordism Tangential stable almost complex structure for a smooth G -manifold M : equivariant isomorphism to a G -complex vector bundle ξ over M : TM × R k ∼ = ξ. Geometric equivariant complex cobordism ring Ω G ∗ . However, Ω G ∗ ≇ π ∗ MU G . The Euler class e V ∈ π − 2 | V | MU G of V is S 0 → S V → Th ( γ | V | G ) . Fact: e V � = 0 if V G = 0. 6 / 24
Geometric equivariant complex cobordism Tangential stable almost complex structure for a smooth G -manifold M : equivariant isomorphism to a G -complex vector bundle ξ over M : TM × R k ∼ = ξ. Geometric equivariant complex cobordism ring Ω G ∗ . However, Ω G ∗ ≇ π ∗ MU G . The Euler class e V ∈ π − 2 | V | MU G of V is S 0 → S V → Th ( γ | V | G ) . Fact: e V � = 0 if V G = 0. 6 / 24
Geometric equivariant complex cobordism Tangential stable almost complex structure for a smooth G -manifold M : equivariant isomorphism to a G -complex vector bundle ξ over M : TM × R k ∼ = ξ. Geometric equivariant complex cobordism ring Ω G ∗ . However, Ω G ∗ ≇ π ∗ MU G . The Euler class e V ∈ π − 2 | V | MU G of V is S 0 → S V → Th ( γ | V | G ) . Fact: e V � = 0 if V G = 0. 6 / 24
Pontryagin-Thom construction Take a cobordant class [ M ] . Equivariant Whiteny’s embedding: M ֒ → V . The normal bundle ν embeds as a tubular neighborhood. Pontryagin-Thom constuction gives a composite map S V → Th ( ν ) → Th ( γ | ν | G ) , which induces a homomorphism Ω G ∗ → π ∗ MU G . The opposite of Thom’s homomorphism does not exist, due to transversality issues. 7 / 24
Pontryagin-Thom construction Take a cobordant class [ M ] . Equivariant Whiteny’s embedding: M ֒ → V . The normal bundle ν embeds as a tubular neighborhood. Pontryagin-Thom constuction gives a composite map S V → Th ( ν ) → Th ( γ | ν | G ) , which induces a homomorphism Ω G ∗ → π ∗ MU G . The opposite of Thom’s homomorphism does not exist, due to transversality issues. 7 / 24

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